Fitness characteristics of ZDV resistant mutants

We used a dataset obtained from the Stanford HIV Drug Resistance Database described in [36] to estimate possible mutational pathways and phenotypic resistance levels under the selective pressure of ZDV. This dataset consists of 1392 observations of HIV reverse transcriptase (RT) genotypes and associated measurements of phenotypic resistance to ZDV. Phenotypic resistance levels are defined as the logarithm of the fold-change in virus' susceptibility to the drug in comparison to the wild type. We focussed on the key thymidine-analog mutations (TAMs) that arise under ZDV monotherapy: 41L, 67N, 70R, 210W, 215Y and 219Q, where, for instance, 41L denotes the presence of leucine (L) at position 41 of the HIV RT. The genotypes considered were classified into those that exclusively contain mutations from the well-studied TAM-1 (41L, 215Y, 210W) or TAM-2 (67N, 70R, 219Q) pathways [37], [38], and two mixed mutant genotypes that contain mutations from both pathways (with cross-TAM profiles). Mixed mutants are observed to generally occur with a lower frequency [39].

Using this dataset, the resistance factors of the genotypes and the partially ordered set (poset) of resistance mutations were estimated using isotonic conjunctive Bayesian network (I-CBN) models. In conjunctive Bayesian networks, a partial order is used to encode dependencies among mutations. A genotype is formally defined as a subset of mutations. The set of genotypes compatible with the order constraints of the poset is called the genotype lattice (Figure 1). In the I-CBN model, isotonic regression is used to associate to each genotype a phenotypic drug resistance level in such a way that resistance levels are non-decreasing along any mutational pathway in the genotype lattice. Next, we used continuous time conjunctive Bayesian networks (CT-CBN) to estimate the rate at which each mutation establishes in the viral population. The fixation times of mutations are assumed to follow independent exponential distributions. The waiting process for a mutation begins only when all its parent mutations in the poset have been established. The data needed for the estimation of this model is a list of genotypes. In the CT-CBN model, genotypes are assumed to be observed after an unknown sampling time. The sampling times are themselves assumed to be random and exponentially distributed. Since we do not know explicitly the time points at which the mutations have occurred, we used the censored CT-CBN model to estimate the fixation rates [40] (See Methods for details).

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Figure 1. Partially ordered set (poset) and induced genotype lattice for mutations associated with resistance to ZDV.

A. Poset of resistance development to ZDV. Vertices represent mutations and directed edges represent the order constraints of mutation accumulation. We observed the clustering of thymidine analog mutations (TAMs) along the two classical TAM-1 and TAM-2 pathways that is well-known under ZDV therapy [37], [38]. The left arm of the poset (mutations 41L, 215Y and 210W) is the TAM-1 pathway, while the right arm (mutations 67N, 70R and 219Q) is the TAM-2 pathway. B. Genotype lattice of mutants induced by the poset of mutations in A. The vertices represent the genotypes that are compatible with the poset in A. Predicted levels of phenotypic resistance are color-coded (green, fully susceptible; red, highly resistant). Please see Supplementary Table S2 in Supporting Information for the waiting times of mutations.

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We used a drug efficacy of on the wild type (corresponding to a drug concentration of 3 times the IC50) to illustrate our results. This value was chosen to match average nadir values in viral load after ZDV monotherapy (a drop of 1 log unit from baseline, within 7–10 days of therapy and a nadir at 3 weeks) [41].

The estimated fitness costs and selective advantages (Table 1 and Figure 2) were in excellent agreement with established knowledge from several in vitro assays and some in vivo observations. The agreement holds true for general reported ranges of fitness costs () [33], [42]–[44] as well as for statements concerning specific mutations. For example, it is well known that the addition of the 210W mutation into a {41L, 215Y} backbone has opposing effects on fitness depending on the presence or absence of ZDV. In the absence of ZDV, the triple mutant {41L, 210W, 215Y} has been observed to be less fit than {41L, 215Y}, while the introduction of ZDV causes a reversal, i.e., the triple mutant becomes fitter than the double mutant [45], [46]. This was well-reflected in our estimates (Table 1). We observed that this reversal in fitness upon adding ZDV, was because of the higher fitness cost of the triple mutant being more than offset by the resistance acquired in the presence of drug. This can be seen by comparing the selective advantages and fitness costs for the corresponding double and triple mutants in the TAM-1 pathway (Figure 2). TAM-1 mutations are known to occur at almost double the frequency of TAM-2 mutations [47]. This difference was reflected in our results by mutants containing TAM-1 mutations having lower fitness costs than those containing TAM-2 mutations. Additionally, we also observed that the selective advantages of TAM-1 mutants is higher, on average, than their TAM-2 counterparts. This is also in concordance with observations that TAM-2 mutations accumulate only after much longer durations of monotherapy with ZDV [39].

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Figure 2. Fitness costs, resistance factors, and selective advantages of mutants arising under ZDV therapy.

A. Estimated fitness costs (normalized by setting fitness cost of wild type to 0), B. Resistance factors (normalized by setting resistance factor of wild type to 1), on a logarithmic scale, and C. Estimated selective advantages (normalized by setting selective advantage of wild type to 1) of ZDV mutants. In A, B and C, the x-axis depicts the number of mutations. The TAM-1 mutants (joined by blue solid lines) are to the left and TAM-2 mutants (joined by red solid lines) are to the right of the wild type. The mixed mutants (joined by black solid lines) are to the left of the TAM-1 mutants. The TAM-1, TAM-2 and mixed mutants are separated by vertical dashed lines.

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Table 1. Estimated fitness costs for ZDV mutants.

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The presence of 41L together with 215Y is a strong predictor of virological failure in patients on ZDV monotherapy [48]. We estimated a low fitness cost for this TAM-1 double mutant and also observed the presence of these two mutations in mutant genotypes contributing to therapy failure.

Our estimated fitness costs were also supported by other in vitro investigations on the order of fitness values, such as the TAM-1 triple mutant {41L, 210W, 215Y} being fitter than its TAM-2 counterpart [43] and the TAM-2 double mutant {67N, 70R} being less fit than the single mutant {67N} [49]. Notably, we concurred with the observation in [49] that the occurrence of 70R in a 67N or {67N, 219Q} backbone has a significant cost.

We further studied parameter identifiability by considering an ensemble of fits. All fits with a root mean squared deviation (RMSD) of between the normalized statistical and mechanistic waiting times, were treated as equally valid (see Methods). Across all such valid fits, we observed a strong and statistically significant Spearman rank correlation of the estimated fitness costs (, p = 0.020) and selective advantages (, p = 0.017). This indicated that our predictions on the ranking of fitness costs and selective advantages of the different mutant genotypes were strongly conserved. Additionally, in about 90% of the valid fits, we found that the average fitness cost of TAM-1 mutants was less than that of TAM-2 mutants, while in approximately of valid fits, the deleterious effect of 210W inserted in a {41L, 215Y} backbone in the absence of ZDV was preserved. Similarly, we examined the validity of each of our conclusions and found that they were all well-conserved across the valid fits (see Supplementary Table S3 and Supplementary Figure S1 in Supporting Information for details). Moreover, since the parameters of the virus dynamics model are subject to uncertainties, we examined the validity of our predictions under uncertainty of all the viral turn-over parameters, considering perturbation of up to . Our model predictions remained robust even under these parameter perturbations (see Supplementary Text S1, Supplementary Figure S2 in Supporting Information for details). Additionally, we also tested the impact of variability of RFs estimated in the first-stage, on the estimation of fitness costs. Again, the order in the estimated fitness costs remained preserved (see Supplementary Text S1 and Supplementary Figure S6).

In addition to estimated fitness characteristics, the viral load time courses predicted by our model gave insights into the dynamics of different mutations. In the TAM-2 pathway, we observed a transient disappearance of mutation 70R before its eventual fixation, as was reported earlier [50]. This phenomenon is attributed to the competition between TAM-1 and TAM-2 mutations: the mutation 70R appears initially and is then outcompeted by 215Y. 70R later fixates in the population after being associated with 67N and other TAM-1 mutations (Figure 3). We also concurred with studies [11] attributing the initial rebound after ZDV monotherapy to insufficient suppression of the wild type, rather than the early selection of mutations.

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Figure 3. Abundance of the 70R mutation and mutant genotypes with 70R under ZDV therapy.

A. Absolute abundance (in numbers) of the 70R mutation. B. Relative abundance of the 70R mutation in the viral population. The transient appearance and eventual fixation of the mutation 70R can be seen. C. Absolute abundance (in numbers) of mutant genotypes containing the mutation 70R. The absolute abundance of a certain mutation is calculated by adding all mutant genotypes containing the mutation.

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Fitness characteristics of IDV resistant mutants

We again used the Stanford HIV Drug Resistance Database [36] to estimate the poset (Figure 4A) and genotype lattice (Figure 4B) of mutations associated with resistance to indinavir (IDV), a protease inhibitor, and corresponding resistance factors of IDV mutants. As for ZDV, the poset, the genotype lattice, and the rate of fixation for each mutation were determined by the I-CBN and CT-CBN models. The dataset for IDV consists of 2170 observations of HIV reverse transcriptase (RT) genotypes and their paired measurements of phenotypic resistance to IDV. We focussed on the five mutations 46I, 54V, 71V, 82A, and 90M. Four of these (46I, 54V, 82A and 90M) are among the most frequent primary (major) mutations reported in the Stanford HIV Drug Resistance Database under IDV therapy [36]. We chose 71V to represent a common secondary (minor) mutation to study possible compensatory fitness effects. We used a drug efficacy on the wild type to illustrate our results. This value was chosen to match average nadir values in viral load after IDV monotherapy (a drop of 1–1.5 log units within 3–4 weeks of therapy) [51], [52].

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Figure 4. Partially ordered set and induced genotype lattice for mutations associated with resistance to IDV.

A. Poset of the continuous time conjunctive Bayesian network for resistance development to IDV. B. The genotype lattice of mutants induced by the poset in A. The vertices represent the genotypes that are compatible with the poset in A. The predicted levels of phenotypic resistance are color-coded (green =  fully susceptible, red  =  highly resistant). Please see Supplementary Table S2 in Supporting Information for the waiting times of mutations.

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The estimated fitness costs, resistance factors and selective advantages (Table 2 and Figure 5) agreed well with reported experimental findings. In general, we observed that early mutations have a high fitness cost, while the accumulation of further mutations succeeded in compensating almost entirely for this loss in fitness (Figure 5A). This is in agreement with clinical observations that mutations selected early during therapy with protease inhibitors cause impaired protease function and that subsequent accumulation of mutations compensates for this fitness cost [27], [53]. A striking behaviour that we observed was the presence of staircases in the fitness landscape, which has also been described earlier [54]. We observed a monotonic increase of the average selective advantages of the mutants with increasing number of mutations (Figure 5C). This observation provides additional reasoning for the accumulation of mutations during IDV therapy. Notably, the high fitness costs for the double and triple mutants (Figure 5A) were not sufficient to deter their occurrence, as the fitness costs were well-offset by resistance (Figure 5B), which facilitated further climbing of the fitness landscape by accumulating mutations (Figure 5C).

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Figure 5. Fitness costs, resistance factors and selective advantages of mutants arising under IDV therapy.

A. Estimated fitness costs (normalized by setting fitness cost of wild type to 0), B. Resistance factors, on a logarithmic scale (normalized by setting resistance factor of wild type to 1), and C. Estimated selective advantages (normalized by setting selective advantage of wild type to 1) of IDV mutants. In A, B and C, the x-axis depicts the number of mutations. Black crosses represent the values for the different mutant genotypes, while the blue solid line represents the average of fitness costs, resistance factors and selective advantages across all mutant genotypes with a given number of mutations.

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Table 2. Estimated fitness costs for IDV mutants.

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In addition to these general fitness trends, specific characteristics of particular mutations were also in line with prior findings. The minor mutation 71V is known to play a compensatory role [55]. In our estimates, this was observed by a partial recovery in fitness of the triple mutant {46I, 71V, 90M} compared to the double mutant {46I, 90M} from 0.66 to 0.44 (Table 2). In the presence of IDV, the addition of 54V to a {71V, 82A} backbone is known to not confer a significant advantage [56], and this was reflected by a ratio of approximately 1.3 for the selective advantages of this pair of mutants. Furthermore, as in [57], we noted a higher fitness cost for the single mutant 71V as compared to 90M.

Nijhuis et al. [55] observed the persistence of protease resistant mutants for long periods of time even after the cessation of therapy. They argued that the reversal of the underlying mutations might not be feasible due to lower replication capacities of intermediate mutants upon reversion. Our results support this hypothesis by showing that the most resistant strain that develops after therapy failure is very unlikely to reverse back in the mutational landscape, owing to a fitness barrier encountered in its reversion to the wild type (Figure 5A).

As with ZDV, we studied parameter identifiability by considering an ensemble of fits with an RMSD between the statistical and mechanistic waiting times. There was a statistically significant Spearman rank correlation (, p = 0.014) between the best estimate of fitnesses and all other valid fits. We found the average fitness estimates (Figure 5C) to be very strongly conserved (, ). We also examined each of the results above and observed consistency across the valid fits (see Supplementary Table S3 and Supplementary Figure S1 in Supporting Information for details). For example, in 77% of the valid fits, 71V was observed to play a compensatory role by lowering fitness costs, while in 65% of fits, the single mutant 90M was fitter than 71V.

Dual therapy with ZDV and IDV

There is great interest in using viral dynamics models to study antiretroviral treatment to assess therapy outcomes and simulate clinical trials [35]. Our model extends naturally to multiple-drug therapy. To illustrate this, we performed simulations of a dual antiretroviral therapy with zidovudine (ZDV) and indinavir (IDV). We used the posets of mutations for ZDV and IDV that we have estimated earlier (Figure 1A and Figure 4A, respectively), together with the resistance factors and fitness costs of the different mutant genotypes (see Methods for details). Our goal was to assess treatment outcomes and reasons for failure of the dual regimen. To this end, we used a range of values for ZDV and IDV to account for differential drug effects and adherence patterns, and studied the treatment outcome by monitoring the total viral load. Our simulations enabled us to predict the dominant mutant genotypes at the point of failure (we defined failure at the point when the viral load crossed a threshold of 500 copies/ml). Based on this, we classified failure as being due to wild type, mutations resistant to ZDV, mutations resistant to IDV or mutations resistant to both drugs used.

We observed that there are different regimes of the individual drug efficacies ( and ) that result in varying causes of failure (Figure 6A). With  = 0.75 and  = 0.90, for example, we observed virological failure after 3 months (Figure 6B) due to mutations resistant to both drugs. In this case, the wild type is sufficiently suppressed and declines during the treatment period. However, we identified regions in the - plane, where treatment failure occurred due to insufficient suppression of the wild type. We classified the treatment as having failed owing to the wild type, if the wild type was the dominant genotype at the point of virological failure. We note that the wild type would eventually be out-competed by resistant mutants in all situations. Such situations of failure with the wild type could indicate insufficient drug pharmacokinetics, a low drug efficacy or poor adherence. This would have implications in designing a salvage therapy regimen, subsequent to failure. Additionally, we also observed that there are combinations of (), for which failure occurs due to mutations to one of the two drugs (Figure 6A). Our model, thus, enabled prediction of viral evolution under a multiple-drug treatment scenario.

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Figure 6. Treatment outcome with ZDV+IDV dual therapy.

A. Genotypic reasons of treatment failure were assessed in terms of mutations present at point of virological failure. For different combinations of drug efficacies and , the different genotypic reasons of failure are shown in different colours. The treatment outcome could be a) failure with mutations resistant to both ZDV and IDV, (b) failure with mutations resistant only to ZDV, (c) failure with mutations resistant only to IDV, (d) failure with wild type, and (e) no detection of failure. B. Viral load (in copies RNA/ml) under ZDV+IDV therapy with  = 0.75 and  = 0.90. The blue line shows the total viral load, while the red dashed line depicts the wild type. The horizontal black dashed line represents the detection threshold used (500 copies/ml).

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We noted that for a certain of one drug, predicting the treatment outcome predicted from monotherapy simulations might lead to qualitatively different results, as opposed to using a model with multi-drug therapy. For example, the value of below which failure with wild type is detected was different in ZDV monotherapy simulations. This reiterates the value of implementing multi-drug treatment regimens in in silico simulations.

We further emphasize that in order to simulate a certain combination, our approach needs only clinical data from treatment regimens in which the individual drugs are a part. For instance, in the current example of dual therapy with ZDV and IDV, the estimation of resistance factors and fitness characteristics relied on sparse cross-sectional clinical data from treatment regimens that included ZDV and/or IDV (not necessarily both). Subsequent to this, we were able to simulate the dual therapy and assess genotypic reasons of therapy failure.

Mechanistic viral dynamics model

The viral infection cycle was described by the two-stage model presented in [34]; see Figure 7 for a graphical representation and description, Supplementary Text S1 for the corresponding system of ordinary differential equations (ODEs) and Supplementary Table S1 for the parameters used. The model allows for integrating drug-specific mutation schemes and the actions of all approved antiretroviral drug classes including reverse transcriptase inhibitors (RTIs), protease inhibitors (PIs) and integrase inhibitors (InIs).

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Figure 7. Two stage mechanistic model of in vivo HIV-1 infection dynamics [6].

Target cells TU (T-cells) and MU (macrophages) can be infected by infective viruses (with effective infection rate constants and ), resulting in early stage infected cells and , respectively. Infection can also be unsuccessful after fusion of the virus, rendering the cell uninfected and thereby eliminating the virus (). and can also possibly return to uninfected states by destruction of essential viral proteins or DNA prior to integration (). cells can enter into a latent state (with probability ) that can get re-activated with a rate constant . Integration of viral DNA in the host genome proceeds with reaction rate constant in the T-cells and in the macrophages, resulting in late stage infected T-cells and macrophages , respectively. The infected cells release new viruses () and non-infective () viruses (with rate constants and , respectively) while the infected cells release new infective and non-infective viruses (with rate constants and , respectively). Target cells TU and MU are produced by the immune system at constant rate with rate constants and , respectively. , , , , and can be cleared by the immune system with reaction rate constants , , , , and , respectively. Viruses are cleared by the immune system with a rate constant . Mutations are modelled to occur at the stage of integration of the viral DNA. The incorporation of the various drug classes is indicated by the inhibition of corresponding processes: EI/FI - entry/fusion inhibitors, NRTI/NNRTI - nucleoside/non-nucleoside reverse transcriptase inhibitors, InI - integrase inhibitors, PI/MI - protease/maturation inhibitors.

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Mutations in the viral genome occur during the process of reverse transcription, but manifest themselves only after the viral DNA has been integrated into the host genome. Hence, mutations were modelled to occur between early infected cells (first stage ) and late infected cells (second stage ). We considered only mutation events between genotypes differing by a single amino acid site, owing to the fact that, though multiple amino acid changes are simultaneously possible, the probability of such events is very low. The probability of a mutation that changes the genotype from to was assumed to be for a single underlying base-pair mutation and for a double underlying base-pair mutation, where denoted the probability of mutation per base-pair per cycle of replication (see Supplementary Text S1 for details, in particular regarding the choice of ). Since mutations are primarily a result of error-prone reverse transcription [70], forward and backward mutations were considered. The average error rate in viral reverse transcription is about mutations per nucleotide per cycle of replication; and two-thirds of these mutations are known to be base-pair substitutions [70]. In agreement with [11], we used the following nucleotide-specific mutation rates: for a G A nucleotide change, we set a mutation rate of (because about half of the base-pair mutations are of this type [70]), for mutations involving nucleotide changes A G, we used a lower rate of , and for transversion and other mutations (involving a change from a purine to a pyrimidine or vice-versa), we set .

As with infectious viruses, we also included different mutant strains of non-infectious viruses, since viral detection assays do not distinguish between infectious and non-infectious viral particles.

The effect of an antiretroviral drug on a viral genotype was modelled by a fractional reduction of the targeted process, characterized by the drug efficacy parameter(1)where denotes the drug concentration at which the fractional reduction is 50%. The subscript indicates that the -value and thus the drug efficacy was assumed to be genotype dependent (see below). Denoting by and the rate constants of the targeted process, in the presence and absence of the drug, respectively, drug action was modelled by(2)The two-stage model in [34] was derived from a more detailed viral infection model by model reduction (see [6] for details). As a consequence, the processes of infection of T-cells and macrophages (with rate constants and , respectively) and production of new infectious and non-infectious viruses from infected T-cells (with rate constants and , respectively) and from infected macrophages (with rate constants and , respectively) are lumped processes, integrating several subprocesses. For example, the infection process comprises the subprocesses of receptor binding, fusion and reverse transcription. The consequences of model reduction have to be taken into account when modelling the actions of drugs targeting some of these subprocesses in the two-stage model, resulting in an additional model reduction factor . This is analogous to the model of competitive inhibition in the context of the Michaelis-Menten approximation of substrate conversion by an enzyme: Defining the unperturbed rate constant as , the model of competitive inhibition stated in terms of the inhibitory concentration in units of would be: with and . For the drug classes of our interest, namely RTIs and PIs, eq. (2) becomes(3)where denotes the probability of successful reverse transcription (for RTIs) or the probability of successful viral maturation (for PIs); and and for RTIs, and and for PIs. For further details, we refer to Supplementary Text of [6].

The advantage of decreased drug-susceptibility of a mutant genotype is typically counter-balanced by a reduction in the fitness of the viral strain [21]. This was quantified in terms of the fitness costs . We made the common assumption that a mutation in a part of the genome that is associated with a certain process of the viral replication cycle caused a drop in the rate of only this process. For example, we assumed that a mutation in the reverse transcriptase part of the HIV-1 genome resulting in a mutant genotype lowers only the rate of reverse transcription. This is a reasonable assumption at least for HIV-1 [71], as most resistance mutations occur in the region of the genome that is coding for the drug target. Hence, any cost due to a resistance mutation is also most likely to be incurred on the function of this region.

Aiming at the integration of in vitro measurements of drug- and strain-specific resistance factors, we parametrized all mutant genotypes with reference to the wild type. This also allowed for the estimation of genotype-specific fitness costs . We denoted by the rate constant of the targeted process in the wild type in the absence of any drug. For a genotype and drug efficacy , the rate constant of the targeted process was defined as(4)The first factor accounts for the reduction in activity due to drug action. The genotype-dependent drug efficacy is given by eq. (1), where(5)was defined in terms of the wild type -value and the resistance factor accounting for the increase in drug-resistance. The second factor in eq. (4) accounts for the reduction in activity due to loss of fitness of the mutant genotype.

In the absence of drug (i.e., ), resistance plays no role and it is clear from eq. (4) that the overall rate of infection for the mutant is lower than that of the wild type, such that the wild type eventually outcompetes the mutant. In the presence of drug, the fitness of a mutant genotype depends on the dynamic interplay between the two factors: the fitness cost and the resistance factor . We quantified the overall fitness of a mutant relative to the wild type based on the selective advantage (similar to [72]) as(6)with by definition. If , the mutant has a replicative advantage over the wild type in the presence of drug.

Statistical model of mutation accumulation

The structure of the mutational landscape determined the scheme of mutations that entered the system of ODEs in the mechanistic model (see Supplementary Text S1). We estimated the mutational scheme from clinical data based on a continuous-time conjunctive Bayesian network (CT-CBN) [16]. The CT-CBN is defined by a partially ordered set (poset) of mutations, denoted by with order relation and by the rate of accumulation (i.e., generation and fixation) of each mutation. The poset specifies the order in which mutations can accumulate in the viral population. The relation indicates that mutation has to occur before mutation . In this case, we call a parent of and denote the set of all parents of by . A subset of mutations is called a genotype. The genotype lattice, denoted by , consists of all genotypes that are compatible with the order constraints of . It defines the subset of possible mutational pathways (see Figure 1 for illustration).

We assume that the occurrence times of mutations follow independent exponential distributions and that the exponential waiting process for a mutation starts only after occurrence of all of its parent mutations in the poset. Formally, for each mutation , we define a random variable . Then we define the statistical waiting times recursively as the random variables(7)for each mutation . In practice, the times of occurrence of mutations are rarely observed and we can solely measure which mutations have been observed at a particular time point, called sampling time or genotyping time. The HIV genome can only be determined if the total viral load is greater than a detection limit  = 500 copies/ml of viral RNA [73]. Hence, in HIV drug resistance development, genotyping can only be performed after therapy failure which is defined as the viral load increasing above the detection limit of a genotypic assay (often 50 copies per mL), thereby precluding the observation of the occurrence of each individual mutation. Sampling times are assumed to be themselves random. In the CT-CBN model, the sampling time is assumed to be exponentially distributed, , and independent of the poset. It is noteworthy that waiting time for the sampling event starts from the onset of therapy (time zero). However, these time points are not available in the Stanford HIV Drug Resistance Database. In this setting, if we observe genotype , then for all and for all .

We assign a resistance factor to each genotype . Resistance factors are typically measured experimentally in vitro by fluorescence-based replication assays [74]. In practice, they are available for some genotypes, but not necessarily for all genotypes of interest. In general, resistance factors observed for a genotype defined by a subset of mutations will vary due to experimental noise and different genetic backgrounds. Supplementary Figure S5 shows the variability of measured resistance factors under ZDV and IDV therapies. To derive a general model, we employ isotonic-CBNs (I-CBNs) and learn a mapping from the genotype space to the drug resistance phenotype space [16]. I-CBN models assume that resistance factors are non-decreasing over the genotype lattice in the direction of evolution. Because of the monotonicity assumption, the regression problem is constrained, which serves as a means of regularization.

The estimation of the statistical models was performed in two steps. In the first step, I-CBN models were estimated separately using 1392 and 2170 in vitro cross-sectional genotype–phenotype observations from the Stanford HIV Drug Resistance Database [36] for ZDV and IDV, respectively. The genotype–phenotype observations were restricted to the [75] or the [76] assays. For ZDV, the I-CBN was applied to mutations 41L, 67N, 70R, 210W, 215Y, and 219Q in the reverse transcriptase region of the viral genome. For IDV, protease mutations 46I, 54V, 71V, 82A, and 90M were analyzed. Each I-CBN model included a poset of mutations and the estimated resistance factors. In the second step, based on the estimated poset, the rate parameters of the CT-CBN models were estimated from the cross-sectional genotype observations of the Stanford HIV Drug Resistance Database using an Expectation-Maximization algorithm [40].

Waiting times, parameter estimation and identifiability

Having specified the parameters of the two-stage dynamical model of viral infection (see Supplementary Table S1 and [6]), a mutation landscape of genotypes, resistance factors and drug efficacies, the only unknown parameters of the model are the fitness costs of the different genotypes. These unknown parameters were estimated by comparison of predicted mechanistic waiting times (based on the mechanistic model of viral infection dynamics) to the statistical waiting times eq. (7). We linked the statistical point of view in terms of mutations and the mechanistic point of view in terms of mutant genotypes as follows.

A given mutant genotype is defined as the set of all mutations that are manifested in . We defined the mechanistic waiting time for each mutation , as the earliest time, at which the following two criteria were satisfied: (i) all the mutant genotypes containing the mutation together constituted at least 20% of the total viral population ; the 20% detection threshold reflected the limitations of the genotyping assay [77]; and (ii) the total viral load was greater than a typical detection limit  = 500 copies/ml of viral RNA, that enables genotyping [73]. This resulted in the following definition of the mechanistic waiting time of mutation :(8)As discussed in the previous subsection, the average waiting times computed from the statistical model are dependent on the sampling rate , to which we typically did not have access. As a solution to this problem, we compared only the relative time scales of appearance of the different mutations. This was done by normalizing the statistical waiting times by the time to the fastest occurring mutation, and then comparing these to correspondingly normalized mechanistic waiting times.

For simulations of the mechanistic viral dynamics model under therapy, we first performed a pre-treatment steady state computation to estimate levels of different mutants at the onset of therapy. All model simulations were performed with MATLAB R2010b. For estimation of fitness costs, we used the MATLAB optimization function fminsearchbnd that is based on the Nelder-Mead simplex direct search algorithm [78] to perform the constrained least-squares estimation(9)where depends on the fitness costs via the system of ODEs specifying the viral dynamics. Fitness costs with were then defined as the solution of eq. (9). Note that in addition to the constraints on fitness costs, the mechanistic waiting times were also subject to the order constraints imposed by the structure of the mutation poset.

In detailed mechanistic models with nonlinearities, parameter identifiability and sloppiness in estimation is a common concern [79], [80]. To study the relevance of this phenomenon in our setting, we performed 500 repeated estimations by choosing different random initial estimates. We also subsequently performed simulated annealing with the function simulannealbnd in MATLAB to test for convergence of our estimates. We did not explicitly observe non-identifiability (indicated by a flat cost function surface in multiple dimensions near the best estimate). However, as in [81], we noted that several rounds of simulated annealing converged to different estimates indicating a rugged search landscape and a possible lack of a well-defined minimum. Following the approach in [81], we considered all fits with an RMSD of less than 0.1 between the mechanistic and statistical waiting times as equally valid (since we estimate the average error in the normalized statistical waiting times data to be ) and discussed our results based on this ensemble of fits.

More details on the RMSD threshold and the number of fits considered are provided in the Supplementary Text S1.

Model simulations for dual therapy

For dual therapy with ZDV and IDV, the poset was constructed by combining the posets of the individual drugs. If and denote the posets under ZDV and IDV monotherapy, then the poset under dual therapy with ZDV and IDV can be written as the disjoint union of the individual posets. That is,(10)with the partial order relation in just being the disjoint union of the partial orders in the individual posets.

If is the genotype lattice induced by the poset under dual therapy, then a mutant genotype has contributions from two sources towards its fitness cost and resistance factor — one from the underlying RT mutations and another due to protease mutations. As before, this simply manifests as changes in reaction rates of appropriate target steps in our viral dynamics model, the only difference being that there are two target steps, instead of one in monotherapy. We assumed mutations in these two regions to be free from fitness and resistance epistasis. This is a reasonable assumption in view of studies indicating that intragenic epistatic effects are more significant than intergenic epistatic effects [28]. We note that if two drugs from the same drug class are to be considered, this either requires some additional assumptions on the extent of epistatic effects or would involve estimation of additional parameters.

For simulations of the dual therapy with ZDV and IDV, we used the variable order ODE-solver ode15s in MATLAB R2010b with relative and absolute tolerances of , and a non-negativity constraint. To study the effects of changing the drug-efficacy parameters and , we performed simulations of the dual therapy for 300 days. To classify therapy outcomes, we monitored total viral load and detected failure when the viral RNA exceed 500 copies/ml. Again, we chose this threshold to correspond with genotyping assay limits. We reported detection of no failure if the viral load did not exceed this threshold within our simulation time.